Properties

Degree $2$
Conductor $3600$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 4·7-s − 2·13-s + 6·17-s + 4·19-s + 6·29-s − 8·31-s − 2·37-s + 6·41-s − 4·43-s + 9·49-s − 6·53-s − 10·61-s − 4·67-s − 2·73-s − 8·79-s − 12·83-s − 18·89-s + 8·91-s − 2·97-s − 18·101-s − 4·103-s + 12·107-s − 10·109-s − 18·113-s − 24·119-s + ⋯
L(s)  = 1  − 1.51·7-s − 0.554·13-s + 1.45·17-s + 0.917·19-s + 1.11·29-s − 1.43·31-s − 0.328·37-s + 0.937·41-s − 0.609·43-s + 9/7·49-s − 0.824·53-s − 1.28·61-s − 0.488·67-s − 0.234·73-s − 0.900·79-s − 1.31·83-s − 1.90·89-s + 0.838·91-s − 0.203·97-s − 1.79·101-s − 0.394·103-s + 1.16·107-s − 0.957·109-s − 1.69·113-s − 2.20·119-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{3600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3600,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.101485244373939343160290559425, −7.35042654382863533673550380313, −6.78354731906038205235867209347, −5.87146842228725166509134997887, −5.34762325653913158633333815914, −4.22720411424127134624171462120, −3.25067910161980170709704604527, −2.84728890251355077691674063068, −1.34557356174820418480074519421, 0, 1.34557356174820418480074519421, 2.84728890251355077691674063068, 3.25067910161980170709704604527, 4.22720411424127134624171462120, 5.34762325653913158633333815914, 5.87146842228725166509134997887, 6.78354731906038205235867209347, 7.35042654382863533673550380313, 8.101485244373939343160290559425

Graph of the $Z$-function along the critical line